An angle bisector ac divides a trapezoid abcd into two similar triangles △abc and △acd. find the perimeter of this trapezoid if the leg ab=9 cm and the leg cd=12 cm.

46 cm

Step-by-step explanation:

It is given that the angle bisector AC divides the trapezoid ABCD into two similar triangles Δ ABC and Δ ACD.

Let us first find the corresponding sides of triangles Δ ABC and Δ ACD.

Since AC is the angle bisector of ∠A,

∠DAC = ∠CAB --- (1)

Also, since ∠DAC and ∠ACB are alternate interior angles,

∠DAC = ∠ACB --- (2)

From (1) and (2),

∠CAB = ∠ACB

Therefore, in Δ ABC,

BC = AB (sides opposite to equal angles are equal)

So, BC = 9 cm.

Now, since ∠DAC = ∠ACB and ∠DAC = ∠CAB, we can take either ∠ACB or ∠CAB as the angle corresponding to ∠DAC. Let us take ∠ACB as the corresponding angle of ∠DAC.

So, the side opposite to ∠DAC in Δ ACD is the corresponding side to the side opposite to ∠ACB in Δ ABC.

That is, the side CD in Δ ACD is the corresponding side to the side AB in Δ ABC.

Now, suppose if the side BC in Δ ABC corresponds to side AD of Δ ACD, then the remaining side AC of Δ ABC should correspond to side AC of Δ ACD which is not possible since they are congruent.

So, the side BC in Δ ABC should correspond to side AC of Δ ACD and the remaining side AC of Δ ABC should correspond to side AD of Δ ACD.

Therefore, we have,

But,

Therefore,

AC = 12 cm

Also,

AD = 16 cm

Now, the perimeter of the trapezoid = AB + BC + CD + AD

= 9 + 9 + 12 + 16

= 46 cm

The perimeter of trapezoid ABCD is 46 cm and this can be determined by using the properties of the triangle.

Given :

An angle bisector AC divides a trapezoid ABCD into two similar triangles △ABC and △ACD.

The leg AB=9 cm and the leg CD=12 cm.

Given that AC is the angle bisector of angle A, therefore:

    ----- (1)

     (alternate interior angles)       ---- (2)

From (1) and (2):

SIdes opposite to equal angles are equal, therefore:

BC = AB

Hence, BC = 9 cm

Side BC in triangle ABC should correspond to side AC of triangle ACD and the remaining side AC of triangle ABC should correspond to side AD of triangle ACD.

Hence,  

AC = 12 cm

AD = 16 cm

Perimeter of Trapezoid = AB + BC + CD + AD

                                       = 9 + 9 + 12 + 16

                                       = 46 cm

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